A Liouville-type theorem for the Lane-Emden equation in a half-space
Analysis of PDEs
2025-04-30 v1
Abstract
We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solution which is monotone in the normal direction. As a consequence, this problem does not admit any positive classical solution which is bounded on finite strips. This question has a long history and our result solves a long-standing open problem. Such a nonexistence result was previously available only for bounded solutions, or under a restriction on the power in the nonlinearity. The result extends to general convex nonlinearities.
Keywords
Cite
@article{arxiv.2003.11466,
title = {A Liouville-type theorem for the Lane-Emden equation in a half-space},
author = {Louis Dupaigne and Boyan Sirakov and Philippe Souplet},
journal= {arXiv preprint arXiv:2003.11466},
year = {2025}
}
Comments
14 pages. arXiv admin note: text overlap with arXiv:2002.07247